Numerical ranges of Foguel operators revisited

Abstract

The Foguel operator is defined as $F_T=\left[\begin{array}{cc}{S}^{\ast }& T\\ 0& S\end{array}\right]$, where $S$ is the right shift on a Hilbert space $\mathcal{H}$ and $T$ can be an arbitrary bounded linear operator acting on $\mathcal{H}$. Obviously, the numerical range $W(F_0)$ of $F_T$ with $T=0$ is the open unit disk, and it was suggested by Gau, Wang and Wu in their LAA'2021 paper that $W(F_{aI})$ for non-zero $a\in \mathbb{C}$ might be an elliptical disk. In this paper, we described $W(F_{aI})$ explicitly and, as it happens, it is not.

9 pages, 2 figures